This is the profile for member infinitely small. This page shows details about infinitely small like postings stats, latest posts, trophies won and interests.
infinitely small

Profile posts Latest activity Postings Post areas Trophies

  • Science Advisor
    Homework Helper
    2020 Award
    Of course almost any set can be given some kind of group structure, but one which ahs nothing at all to do with its topological structure. E.g. any finite set can be made into a cyclic group and any countable set can be made into the integer group, and any set with cardinality equal to that of the reals can be made into the reals as a group.
    infinitely small
    infinitely small
    What is the proof for the almost any set can be given some kind of group structure?Proofs for the others? Because i am interested in those proofs.Thank you.
    M
    mathwonk
    Science Advisor
    Homework Helper
    2020 Award
    if S is a set of n elements, that means there is a bijection from S to the set Z/n = {0,1,2,....,n-1}. Since that set has a group structure as a cyclic group, namely integers modulo n, you define an isomorphic group structure on the set S by first fixing a bijection f:S-->Z/n, and then taking any two elements of S, you add them by going over to Z/n by the bijection f, adding them there, and then bringing their sum back by the inverse of the bijection f to get an element of S.
    M
    mathwonk
    Science Advisor
    Homework Helper
    2020 Award
    Similarly any set that has a bijection to an group, cvan be made into a group in this way. Hnce any set with a bijection to the ral numbers, can be made into a group isomorphic to the additive goup of R. If this is puzzling, you would benefit from reading some basic set theory, about bijections and so on.
    Science Advisor
    Homework Helper
    2020 Award
    mr small, the key geometric property of groups that are also manifolds, is that they have trivial tangent bundle. Inparticular they admit a vector field without zeroes. Hence the euler characteristic must be zero. In the case of surfaces this shows that the genus must be one, the only case of a smooth surfce which also has a (smooth) group structure.
  • Loading…
  • Loading…
  • Loading…
  • Loading…
  • Loading…

Find Member

Top